Optimal Consumption and Investment Strategies with a Perishable and an Indivisible Durable Consumption Good

We study the consumption and investment choice of an agent in a continuous-time economy with a riskless asset, several risky financial assets, and two consumption goods, namely a perishable and a durable good with an uncertain price evolution. Assuming lognormal prices and a multiplicatively separable, isoelastic utility function, we provide an explicit Merton-type solution for the optimal strategies for the case where the durable (and all other assets) can be traded without transaction costs. For the case where the durable good is indivisible, in the sense that durable trades imply transaction costs proportional to the value of the current durable holdings, we show analytically that the optimal durable trading strategy is characterized by three constants \underline{z} lt; z* lt; \overline{z}. As long as the ratio z of the total current wealth to the value of current durable holdings of the investor is in (\underline{z},\overline{z}), it is optimal not to trade the durable. At the boundaries of this interval it is optimal to trade the durable to attain z = z*. The model is used to examine the optimal substitution between perishable and durable consumption and the importance of the durable price uncertainty and the correlation between the price of the durable good and financial asset prices